Lesson 2.5, page 312
Zeros of Polynomial Functions
Objective: To find a polynomial
with specified zeros, rational
zeros, and other zeros, and to use
Descartes’ rule.
Introduction
Polynomial
5x3 + 3x2 + (2 + 4i) + i
Type of Coefficient
complex
5x3 + 3x2 + √2x – π
real
5x3 + 3x2 + ½ x – ⅜
rational
5x3 + 3x2 + 8x – 11
integer *******
Rational Zero Theorem
If the polynomial
f(x) = anxn + an-1xn-1 + . . . + a1x + a0
has integer coefficients, then every rational
zero of f(x) is of the form
p
q
where p is a factor of the constant a0
and q is a factor of the leading coefficient an.
Rational Root (Zero) Theorem
• If “q” is the leading coefficient and “p” is the
constant term of a polynomial, then the only
possible rational roots are + factors of “p” divided
by + factors of “q”. (p / q)
5
3
f
(
x
)

6
x

4
x
 12 x  4
• Example:
• To find the POSSIBLE rational roots of f(x), we
need the FACTORS of the leading coefficient (6 for
this example) and the factors of the constant term
(4, for this example). Possible rational roots are
 factors of p
1, 2, 4
1 1 1 2 4


  1, 2, 4, , , , , 
 factors of q 1, 2, 3, 6
2 3 6 3 3

See Example 1, page 313.
• Check Point 1: List all possible rational
zeros of f(x) = x3 + 2x2 – 5x – 6.
Another example
• Check Point 2: List all possible rational
zeros of f(x) = 4x5 + 12x4 – x – 3.
How do we know which possibilities
are really zeros (solutions)?
• Use trial and error and synthetic division to
see if one of the possible zeros is actually
a zero.
• Remember: When dividing by x – c, if the
remainder is 0 when using synthetic
division, then c is a zero of the polynomial.
• If c is a zero, then solve the polynomial
resulting from the synthetic division to find
the other zeros.
See Example 3, page 315.
• Check Point 3: Find all zeros of
f(x) = x3 + 8x2 + 11x – 20.
Finding the Rational Zeros of a Polynomial
1. List all possible rational zeros of the
polynomial using the Rational Zero Theorem.
2. Use synthetic division on each possible
rational zero and the polynomial until one
gives a remainder of zero. This means you
have found a zero, as well as a factor.
3. Write the polynomial as the product of this
factor and the quotient.
4. Repeat procedure on the quotient until the
quotient is quadratic.
5. Once the quotient is quadratic, factor or use
the quadratic formula to find the remaining
real and imaginary zeros.
Check Point 4, page 316
• Find all zeros of
f(x) = x3 + x2 - 5x – 2.
How many zeros does a polynomial
with rational coefficients have?
• An nth degree polynomial has a total of n zeros.
Some may be rational, irrational or complex.
• Because all coefficients are RATIONAL, irrational
roots exist in pairs (both the irrational # and its
conjugate). Complex roots also exist in pairs (both
the complex # and its conjugate).
• If a + bi is a root, a – bi is a root
• If a  b is a root, a  b is a root.
• NOTE: Sometimes it is helpful to graph the
function and find the x-intercepts (zeros) to
narrow down all the possible zeros.
See Example 5, page 317.
• Check Point 5
• Solve: x4 – 6x3 + 22x2 - 30x + 13 = 0.
Fundamental Theorem of Algebra
(page 318)

• If f(x) is a polynomial function of degree
n, where n > 1, then the equation f(x) =
0 has at least one complex zero.
• Note: This theorem just guarantees a
zero exists, but does not tell us how to
find it.
Linear Factorization Theorem, pg.
319
Remember…
• Complex zeros come in pairs as
complex conjugates: a + bi, a – bi
• Irrational zeros come in pairs.
a c b , a c b
Practice
Find a polynomial function
(in factored form) of degree 3
with 2 and i as zeros.
More Practice
Find a polynomial function
(in factored form) of degree 5
with -1/2 as a zero with multiplicity 2,
0 as a zero of multiplicity 1,
and 1 as a zero of multiplicity 2.
Example
• Find a 4th-degree polynomial function f(x)
with real coefficients that has -2, 2 and i as
zeros and such that f(3) = -150.
Extra Example
Suppose that a polynomial function of
degree 4 with rational coefficients has i
and -3 +√3 as zeros. Find the other
zero(s).